where is the frequency of a succesful `hop' and is the
average hop-distance.

The reaction rate theory was worked out for the case of activation
over a particular barrier, i.e.,

(16-2)

The relation of
to the total succesful hop frequency
is the sum of hop frequencies of all surrounding sites

(16-3)

where
is the particular hop frequency into a site of type
and is the degeneracy (i.e., count) of similar neighbors.

For example, consider the case of an interstitial atom in FCC-iron.
There are four crystallographically similar nearest interstitial sites (
);
if the successful hop frequency into the next nearest neighbor,
, is
negligible compared to the nearest neighbor hops (
)
then

(16-4)

writing the hop distance in terms of the lattice constant , and combining
with equations 16-1 and 16-4 for uncorrelated hops (), yields

(16-5)

as a good approximation that connects atomistic mechanisms to macroscopic diffusivity for interstitial
diffusion in a hydrostatically stressed FCC crystal.

The Gibbs free energy of the activated state in the exponent results from the caveat of fixed stress.
The Helmholtz free energy would appear for the case of fixed volume.
Hydrostatic stress is results from the assumption that each neighboring interstitial site is
the same--this would not be the case, for example, in tension
(
and
).
Tension would break the symmetry of the interstitial sites by a tetrahedral distortion.
This effect has been observed in steel.

Simple Models for Vacancy Diffusivity

The transition of an atom into a vacant site has the same treatment as the above
case for interstitials, except that two additional factors must be considered:

Vacancy Population

The probability of an atom passing through its activated state and into a vacant site
is the product of two probabilities: 1) the activation probability,
, and
2) the probability that the neighboring site will be vacant, , the site fraction
of vacant sites.

The activation barrier, , depends on the presence of a neighboring vacancy.
The terms in the probabilities are not necessarily independent, but it is
a good approximation to treat the probabilities as independent terms so that
they multiply.

Correlations

Because an atom that has just hopped into a vacant site will
always have a vacancy as a neighbor (i.e., the site that it had previously
occupied), there is a strong correlation between steps and thus as
in an uncorrelated random walk.

With the physically justifiable assumption that vacant site occupation is small (thus
neglecting the probability of two neighboring vacancies1, the successful hop frequency can be related to the
sum of particular vacancy hops as in equation 16-3.

For self-diffusion on a hydrostatically stressed FCC lattice, there are 12 crystallographically
similar nearest neighbor sites,
.
Therefore the successful hop frequency of an FCC-vacancy is:

(16-6)

where is the Gibbs free energy of a the vacancy in its
activated transition state.

Disregarding all other mechanisms of diffusion (i.e., interstiticialcy)
The successful hop frequency, of an atom chosen at random
is the product of two probabilities:

(16-7)

If the vacancies are in equilibrium, can be related to the
Gibbs free energy difference between an occupied and unoccupied site,
, and2

(16-8)

Collecting terms and relating the jump distance, , to the lattice constant
,

(16-9)

The correlation factor, , can be estimated with the following model.
A specific atom is likely hop back into its original site
times--therefore
a vacancy will typically have nonzero net displacement after about
attempts,
where two of the attempts (the back-and-forth steps) will have been ineffective.
Therefore only fraction,
, of all hops should be considered
succesful.
For an FCC lattice,
.
This approximation is not very different those obtained from
more accurate models.

Lattice Diffusion in Ionic Crystals

Just as in substitutional diffusion via the vacancy mechanism in which
the diffusivity has an Arrhenius factor associated with the population
of defects, a similar defect population appears in ionic crystals
where the defects are more complicated.

In an ionic crystal, a vacancy at a cation (positive ion) or an anion
site would leave a charge of the opposite sign unless that vacancy is
compensated by another charged defect.
The two common types of defects is a Schottky defect, where a pair of
vacancies of opposite signs appear on each sublattice--and a Frenckel
defect where a vacancy is compensated by an interstitial.
If the cations and anions have differing magnitudes of ionic charge,
then the defects must involve more sites to satisfy charge neutrality.

In ionic crystals, the notation for defect concentration identifies
the chemical species, where the species is located relative to a
perfect lattice, and the charge on the species.
The commonly employed notation for charged defects is Kroger-Vink:

where

(16-10)

where,

(16-11)

For example consider, putting calcia into a zirconia lattice:

CaO

(16-12)

The Schottky formation reaction is given by:

The equilibrium for Shottkey formation involves two unknowns:

Charge neutrality provides an additional constraint:

This forms a set of equations that can be solved for the
defect concentration that affects the diffusivity.